Right Triangle Trigonometric Functions

Definition of Trigonometric Functions in Right Triangles

Trigonometric functions are defined using the ratios of the sides of a right triangle. For an acute angle θ in a right triangle with sides labeled as opp (opposite), adj (adjacent), and hyp (hypotenuse), the six trigonometric functions are:

• Sine:

• Cosine:

• Tangent:

• Cosecant:

• Secant:

• Cotangent:

Example: In a right triangle with sides 3 (opp), 4 (adj), and 5 (hyp), , , .

Simplifying Radical Expressions

Process and Examples

Radical expressions often need to be simplified and rationalized. The table below demonstrates common simplifications:

Key Point: Rationalizing the denominator means rewriting the expression so that no radicals appear in the denominator.

Special Right Triangles and Trigonometric Values

45°-45°-90° and 30°-60°-90° Triangles

Special right triangles have side ratios that allow for easy calculation of trigonometric values for common angles.

• 45°-45°-90° Triangle: Both legs are 1, hypotenuse is .

• 30°-60°-90° Triangle: Shorter leg is 1, longer leg is , hypotenuse is 2.

Trigonometric Functions for Acute Angles:

Example: In a 30°-60°-90° triangle, , .

Trigonometric Identities

Quotient, Reciprocal, and Pythagorean Identities

• Quotient Identities:

• Reciprocal Identities:

• Pythagorean Identities:

Example: If , then .

Co-Function Identities and Complementary Angles

Relationships Between Trigonometric Functions of Complementary Angles

Co-function identities relate the trigonometric functions of complementary angles (angles that add up to radians or 90°):

Key Point: The acute angles of every right triangle are complementary.

Calculator Mode Warning

Caution: Always ensure your calculator is set to the correct mode (degrees or radians) when evaluating trigonometric functions. Using the wrong mode will result in incorrect answers.